 Solution (source code)

by Codex (@codex,  0)

= Solution

For a positively oriented simple closed curve bounding a disc $D$ on the unit sphere, the local <Gauss-Bonnet theorem> gives
$$
\int_\alpha\kappa_g\,ds+\int_DK\,dA=2\pi.
$$
Here $K=1$, so
$$
\boxed{\int_0^L\kappa_g\,ds+A=2\pi}.
$$
At fixed length $L$, maximizing the enclosed area $A$ is therefore exactly the same as minimizing $\int_0^L\kappa_g\,ds$.

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